Dominance analysis (DA) determines the relative importance of independent variables as well as parameter estimates in an estimation model based on contribution to an overall model fit statistic (see Grömping, 2007 as well as Luchman, Lei, and Kaplan, 2020 for discussion). DA is an ensemble method in which importance determinations about independent variables or parameter estimates are made by aggregating results across multiple models, though the method usually requires the ensemble contain each possible combination of the independent variables in the full model.
The all possible combinations ensemble with p independent variables in the full model results in 2^p-1 models estimated. That is, each combiation of p variables alterating between included versus excluded (i.e., the 2 base to the exponent) where the constant[s]-only model is omitted (i.e., the -1 representing the distinct combination where no independent variables are included; see Budescu, 1993).
domme is implemented as a flexible wrapper command that can be used with most Stata estimation commands that use Stata's ml estimation and allow constraints(). domme can produce its own fit statistics for commands that return the e(ll) log-likelihood scalar.
domme is an extension of domin and readers are referred to the domin page for basics on dominance analysis.
Some examples of the command as applied to Stata estimation commands are shown below.
Seemingly unrelated regression (i.e., sureg) is an analysis that cannot be accommodated with the domin command but can easily be accommodated by domme.
An example sureg analysis is reported below.
. sysuse auto
(1978 Automobile Data)
. sureg (price = length foreign gear_ratio) (headroom = mpg)
Seemingly unrelated regression
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Equation Obs Parms RMSE "R-sq" chi2 P
--------------------------------------------------------------------------
price 74 3 2319.086 0.3733 47.80 0.0000
headroom 74 1 .764981 0.1712 16.18 0.0001
--------------------------------------------------------------------------
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
price |
length | 71.43614 16.93133 4.22 0.000 38.25134 104.6209
foreign | 3903.345 826.7769 4.72 0.000 2282.892 5523.798
gear_ratio | -2512.173 954.5322 -2.63 0.008 -4383.022 -641.3239
_cons | -846.5025 5212.856 -0.16 0.871 -11063.51 9370.507
-------------+----------------------------------------------------------------
headroom |
mpg | -.0618228 .0153694 -4.02 0.000 -.0919463 -.0316992
_cons | 4.309902 .3391911 12.71 0.000 3.645099 4.974704
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sureg does not produce its own fit statistic but does record the e(ll) scalar. In the example below, fitstat(e(), mcf) is a built-in option that computes a fit statistic to dominance analyze. The dominance results for the sureg command are reported below. Note that the focus of domme is on parameter estimate importance/PERI and not the importance of independent variables.
. domme (price = length foreign gear_ratio) (headroom = mpg), reg(sureg (price = length foreign gear_ratio) (headroom = mpg)) fitstat(e(), mcf)
Total of 15 models/regressions
General dominance statistics: sureg
Number of obs = 74
Overall Fit Statistic = 0.0322
| Dominance Standardized Ranking
| Stat. Domin. Stat.
------------+------------------------------------------------------------------------
price |
length | 0.0109 0.3375 1
foreign | 0.0070 0.2169 3
gear_ratio | 0.0051 0.1576 4
headroom |
mpg | 0.0093 0.2881 2
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Conditional dominance statistics
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#param_ests: #param_ests: #param_ests: #param_ests:
1 2 3 4
price:length 0.0100 0.0116 0.0118 0.0101
price:foreign 0.0004 0.0056 0.0097 0.0124
price:gear_ratio 0.0042 0.0059 0.0060 0.0042
headroom:mpg 0.0087 0.0093 0.0097 0.0094
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Complete dominance designation
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dominated?: dominated?: dominated?: dominated?:
length foreign gear_ratio mpg
dominates?:length 0 0 1 0
dominates?:foreign 0 0 0 0
dominates?:gear_ratio -1 0 0 0
dominates?:mpg 0 0 0 0
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Strongest dominance designations
price:length completely dominates price:gear_ratio
headroom:mpg conditionally dominates price:gear_ratio
price:length conditionally dominates headroom:mpg
price:length generally dominates price:foreign
headroom:mpg generally dominates price:foreign
price:foreign generally dominates price:gear_ratio
Note that any parameters not included in the dominance analysis are considered to be a part of the constant model for built-in fit statistics like e(mcf).
A zero inflated Poisson (i.e., zip) is another analysis that can be accommodated by domme
. generate zi_pr = price*foreign
. domme (zi_pr = headroom trunk) (inflate = gear_ratio turn), reg(zip zi_pr headroom trunk) f(e(), bic) ropt(inflate(gear_ratio turn)) reverse
Total of 15 models/regressions
General dominance statistics: Zero-inflated Poisson regression
Number of obs = 74
Overall Fit Statistic = 18246.3488
Constant-only Fit Stat. = 20760.9084
| Dominance Standardized Ranking
| Stat. Domin. Stat.
------------+------------------------------------------------------------------------
zi_pr |
headroom | -124.0851 0.0493 2
trunk | -2344.9819 0.9326 1
inflate |
gear_ratio | -26.9970 0.0107 3
turn | -18.4955 0.0074 4
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Conditional dominance statistics
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#param_ests: #param_ests: #param_ests: #param_ests:
1 2 3 4
zi_pr:headroom -248.5252 -165.5652 -82.6051 0.3549
zi_pr:trunk -2469.4220 -2386.4619 -2303.5019 -2220.5418
inflate:gear_ratio -41.6540 -31.8827 -22.1114 -12.3401
inflate:turn -33.1525 -23.3812 -13.6099 -3.8386
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Complete dominance designation
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dominated?: dominated?: dominated?: dominated?:
headroom trunk gear_ratio turn
dominates?:headroom 0 -1 0 0
dominates?:trunk 1 0 1 1
dominates?:gear_ratio 0 -1 0 1
dominates?:turn 0 -1 -1 0
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Strongest dominance designations
zi_pr:trunk completely dominates zi_pr:headroom
zi_pr:trunk completely dominates inflate:gear_ratio
zi_pr:trunk completely dominates inflate:turn
inflate:gear_ratio completely dominates inflate:turn
zi_pr:headroom generally dominates inflate:gear_ratio
zi_pr:headroom generally dominates inflate:turn